An Algebraic Characterization of singular quasi-bi-Hamiltonian systems

TL;DR

This paper provides an algebraic criterion to characterize singular quasi-bi-Hamiltonian systems, broadening the understanding of their structure and showing that non-singular Poisson tensors are not necessary, with illustrative examples.

Contribution

It introduces a new algebraic criterion for singular quasi-bi-Hamiltonian systems and generalizes existing methods for constructing Poisson tensors.

Findings

Established an algebraic criterion for singular quasi-bi-Hamiltonian systems

Demonstrated that non-singular Poisson tensors are not essential for these systems

Connected existing Poisson tensor construction methods to their approach

Abstract

In this paper we prove an algebraic criterion which characterizes singular quasi-bi-hamiltonian structures constructed on the lines of a general, simple, new formal procedure proposed by the authors. This procedure shows that for the definition of a quasi-bi-hamiltonian system the requirement of non-singular Poisson tensors, contained in the original definition by Brouzet et al., is not essential. Besides, it is incidentally shown that one method of constructing Poisson tensors available in the literature is a particular case of ours. We present 2 examples.